Sparse Parity-Check Matrices over ${GF(q)}$

• Authors: Hanno Lefmann
• Venue: Combinatorics, Probability and Computing
• Year: 2005

For fixed positive integers $k,q,r$ with $q$ a prime power and large $m$, we investigate matrices with $m$ rows and a maximum number $N_q (m,k,r)$ of columns, such that each column contains at most $r$ nonzero entries from the finite field $GF(q)$ and any $k$ columns are linearly independent over $GF(q)$. For even integers $k \geq 2$ we obtain the lower bounds $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))})$, and $N_q(m,k,r) = \Omega (m^{((k-1)r)/(2(k-2))})$ for odd $k \geq 3$. For $k=2^i$ we show that $N_q(m,k,r) = \Theta ( m^{kr/(2(k-1))})$ if $\gcd(k-1,r) = k-1$, while for arbitrary even $k \geq 4$ with $\gcd(k-1,r) =1$ we have $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))} \cdot (\log m)^{1/(k-1)})$. Matrices which fulfil these lower bounds can be found in polynomial time. Moreover, for $\Char (GF(q)) 2 $ we obtain $N_q(m,4,r) = \Theta (m^{\lceil 4r/3\rceil/2})$, while for $\Char (GF(q)) = 2$ we can only show that $N_q(m,4,r) = O (m^{\lceil 4r/3\rceil/2})$. Our results extend and complement earlier results from [7, 18], where the case $q=2$ was considered.